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The Gaussian mechanism is an essential building block used in multitude ofdifferentially private data analysis algorithms. In this paper we revisit theGaussian mechanism and show that the original analysis has several importantlimitations. Our analysis reveals that the variance formula for the originalmechanism is far from tight in the high privacy regime ($\varepsilon \to 0$)and it cannot be extended to the low privacy regime ($\varepsilon \to \infty$).We address these limitations by developing an optimal Gaussian mechanism whosevariance is calibrated directly using the Gaussian cumulative density functioninstead of a tail bound approximation. We also propose to equip the Gaussianmechanism with a post-processing step based on adaptive estimation techniquesby leveraging that the distribution of the perturbation is known. Ourexperiments show that analytical calibration removes at least a third of thevariance of the noise compared to the classical Gaussian mechanism, and thatdenoising dramatically improves the accuracy of the Gaussian mechanism in thehigh-dimensional regime.

Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising